Metric spheres in the projective spaces with constant holomorphic sectional curvature (Q638748)

Let \(\widetilde{M}^n(c)=\mathbb{K}P^k(c) \) be the projective space of constant holomorphic sectional curvature \(c>0\). Here, \(\mathbb{K}=\mathbb{C}\) for \(\lambda =1\), \(\mathbb{K}=\mathbb{Q}\) for \(\lambda =3\) and \(\mathbb{K}=\mathbb{C}a\) for \(\lambda =7\) and \(k=2\) where the real dimension \(n\) is \(n=(\lambda +1)k\) with \(k\geq 2\). The purpose of this article is to establish rigidity results for metric spheres in \(\widetilde{M}^n(c)\). In order to avoid a case-by-case discussion on the choice of coefficient fields, the authors use the matrix Jacobi equation along the unit speed geodesics fitting a fixed unit normal field.